We note that if \({\mathbf {x}}_i\)  is mean-zero and nSG(\(\sigma _i\) ), then by lemma:nSGcomponentSG, \(*{{\mathbf {u}}_i,{\mathbf {x}}_i} \mid {\mathcal {F}}_{i-1}\)  is
zero-mean and \(*{{\mathbf {u}}_i} \sigma _i\) -subGaussian. The rest of the proof follows from the proof of Lemma 39 in [1]} (key idea is exponentiate and then apply Markov's inequality). For completeness, we restate the proof here. Observe that for any \(i\) , since \(*{{\mathbf {u}}_i,{\mathbf {x}}_i}\)  is \(*{{\mathbf {u}}_i} \sigma _i\) -subGaussian, for any \(\lambda > 0\) , we have that
\(\mathbb {E}\left[\exp (\lambda *{{\mathbf {u}}_i, {\mathbf {x}}_i}) \mid {\mathcal {F}}_{i-1} \right] \le \exp (\lambda ^2 *{{\mathbf {u}}_i}^2 \sigma _i^2/2)\) 
